Acute Angle

Introduction

An acute angle is a kind of angle that is less than `90°`. For instance, when the clock shows `7` o'clock, the angle formed between the hour hand and the minute hand is an acute angle. It's like the angle when you bend your elbow, making a sharp but not wide opening. This type of angle falls within `0°` to `90°`. Acute angles are crucial in making different shapes, especially triangles, and are useful in various math and science situations.

Understanding Acute Angles

An angle happens when two lines meet at a point. If that angle is smaller than `90` degrees, we call it an acute angle. Some examples of acute angles are `30^\circ`, `50^\circ`, and `75^\circ`. These kinds of angles show up a lot in shapes like triangles, and they're useful in lots of areas of maths and science.

An acute angle is like a tiny corner where two edges meet, forming a sharp, narrow angle that's smaller than `90^\circ`. Picture it as if you're looking at a slice of cake or a piece of pie—the angle of the arc is pointed and not wide. 

• The vertex is the point where the two lines of the angle meet, similar to the tip of the slice in the cake analogy. 
   
• The lines extending from the vertex, called the arms of the angle, are like the two sides of the cake slice. They spread apart but not too much, always staying smaller than a right angle, which is `90^\circ`.
   

Acute Angle Properties

The basic properties of acute angles are:

• It must be greater than zero, \(0^\circ < \theta\).
• It must be less than a right angle, \(90^\circ > \theta\).
• The sine of any acute angle falls between `0` and `1`.
• The cosine of any acute angle falls between `1` and `0`.
• The tangent of any acute angle falls between `0` and infinity.

Acute Angles in Real World 

`1`. Architectural Design: Architects use acute angles to design aesthetically pleasing structures, such as sloped roofs or angular facades, which rely on acute angles to achieve balance and visual appeal.

`2`. Engineering Mechanics: Engineers utilize acute angles in designing mechanical components, such as gear teeth or cam profiles, where precise angles are necessary for efficient operation and motion control.

`3`. Computer Graphics: Acute angles are crucial in computer graphics for rendering realistic images and animations. They determine the viewing perspective and angles of objects in virtual environments, contributing to the realism of simulations and video games.

`4`. Trigonometry and Mathematics: Acute angles are fundamental in trigonometry, forming the basis of trigonometric functions such as sine, cosine, and tangent. These functions are extensively used in fields like physics, engineering, and computer graphics for modeling various phenomena and calculations involving angles and distances.

`5`. Navigation: Acute angles are essential in navigation systems, such as GPS, where they help calculate the direction and distance between locations. They also play a role in determining optimal routes for transportation, such as flight paths for aircraft or navigation routes for ships.

Properties of Acute Angled Triangles

An acute triangle is a type of triangle where all its angles are less than `90^\circ`. Now, when all three angles of a triangle are exactly `60^\circ` , we call it an equilateral triangle. Think of it as a special case of an acute angle triangle where all the angles are the same, like having three identical corners. 

Acute triangles can be further divided into acute scalene triangles (where all three sides have different lengths), acute isosceles triangles (where two sides have the same length), and equilateral triangles. So, basically, acute triangles are a type of triangle with all interior angles less than `90^\circ`.

Acute Angle Formula

Just like Pythagoras' theorem deals with right triangles, there's a similar rule for acute angle triangles known as the triangle inequality theorem. This theorem states that in any acute angle triangle, the sum of the squares of the two shorter sides is always greater than the square of the longest side. 

So, if you have a triangle `ABC` with sides `a`, `b`, and `c` (where `c` is the longest side), then `a^2 + b^2 > c^2`. In simple terms, if `a^2 + b^2` is greater than `c^2`, you're dealing with an acute angle triangle. This rule helps us understand how the sides of an acute triangle relate to each other. 

• \(a^2 + b^2 > c^2\)
• \(c^2 + b^2 > a^2\)
• \(a^2 + c^2 > b^2\)

Solved Examples

Example `1`: Does the given triangle form an acute triangle?

Solution:

In an acute triangle, all three angles must be less than \(90^\circ \). Looking at the figure, we observe that two angles measure \(30^\circ \) each and \(60^\circ \), and \(\angle R \) is \(90^\circ \). Thus, this triangle does not meet the criteria for an acute triangle; instead, it forms a right-angled triangle.

Example `2`: Determine which of the following angles are acute:

\( \angle 90^\circ, \angle 47^\circ, \angle 123^\circ, \angle 0^\circ, \angle 312^\circ, \angle 25^\circ \)

Solution:

We understand that for an angle to be considered acute, it must fall within the range of \(0^\circ\) to \(90^\circ\).

From the provided list of angles, only \(\angle 47^\circ\) and \(\angle 25^\circ\) fall within this specified range.

Therefore, \(\angle 47^\circ\) and \(\angle 25^\circ\) are the only acute angles.

Example `3`: Given a triangle with side lengths \( a = 5 \, \text{cm} \), \( b = 12 \, \text{cm} \), and \( c = 13 \, \text{cm} \), is this triangle acute?

Solution:  

Given: \( a = 5 \, \text{cm} \), \( b = 12 \, \text{cm} \), and \( c = 13 \, \text{cm} \).

Using the triangle inequality theorem, we have:  

 \(x^2 + y^2 > z^2\)  

\(5^2 + 12^2 > 13^2\)  

\(25 + 144 > 169\)  

\(169 > 169\)  

Since \(169 > 169\) is not a true statement, the triangle is not acute.

Example `4`: Given the angles of a triangle as \( \angle 33^\circ \), \( \angle 49^\circ \), and \( \angle 98^\circ \), does this triangle qualify as an acute triangle?

Solution: 

We observe that in order for a triangle to be classified as acute, all three angles must fall between \(0^\circ\) and \(90^\circ\). However, in this case, we have one angle measuring \(98^\circ\).

Therefore, this triangle does not meet the criteria for an acute triangle; instead, it qualifies as a obtuse triangle due to one of its angles being \(99^\circ\).

Example `5`: Given a triangle with side lengths \( x = 7 \, \text{m} \), \( y = 9 \, \text{m} \), and \( z = 10 \, \text{m} \), is this triangle acute?

Solution:  

Given: \( x = 7 \, \text{m} \), \( y = 9 \, \text{m} \), and \( z = 10 \, \text{m} \).

Using the triangle inequality theorem, we have:  

\(x^2 + y^2 > z^2\)  

\(7^2 + 9^2 > 10^2\)  

\(49 + 81 > 100\)  

\(130 > 100\)  

Since \(130 > 100\) is true statement, the triangle is acute.

Practice Problems

Q`1`. Determine which of the following angles is acute:

\( \angle 56^\circ, \angle 90^\circ, \angle 245^\circ, \angle 0^\circ, \angle 133^\circ, \angle 38^\circ \)

1. \( \angle 245^\circ \) and \( \angle 0^\circ\)
2. \( \angle 56^\circ \) and \( \angle 38^\circ\)
3. \( \angle 90^\circ \) and \( \angle 133^\circ\)
4. \( \angle 38^\circ \) and \( \angle 245^\circ\)

Answer: b

Q`2`. Given a triangle with side lengths \( a = 3 \, \text{cm} \), \( b = 4 \, \text{cm} \), and \( c = 5 \, \text{cm} \), is this triangle acute?

1. Yes
2. No

Answer: b

Q`3`. Does the given triangle form an acute triangle?

1. Yes
2. No

Answer: a

Q`4`. Identify the acute angle from the given options:

1. \( \angle 150^\circ \)
2. \( \angle 55^\circ \)
3. \( \angle 100^\circ \)
4. \( \angle 95^\circ \)

Answer: b

Frequently Asked Questions

Q`1`. What is the Pythagorean theorem?

Answer: The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (\(c\)) is equal to the sum of the squares of the lengths of the other two sides (\(a\) and \(b\)). Mathematically, it is represented as \(a^2 + b^2 = c^2\).

Q`2`. What is the Triangle Inequality Theorem for acute angle triangles?

Answer: The Triangle Inequality Theorem states that in any acute triangle, the sum of the squares of the two shorter sides is always greater than the square of the longest side. In mathematical terms, for a triangle with sides of lengths \(a\), \(b\), and \(c\), this can be expressed as \(a^2 + b^2 > c^2\), where `c` is the longest side.

Q`3`. Define an acute triangle.

Answer: An acute triangle is a triangle in which all three angles measure less than `90` degrees (\(90^\circ\)). In mathematical notation, for angles \(A\), \(B\), and \(C\) in triangle \(ABC\), this can be expressed as \(\angle A < 90^\circ\), \(\angle B < 90^\circ\), and \(\angle C < 90^\circ\).

Q`4`. What is the range of acute angles?

Answer: The range of acute angles is from `0°` to `90°`, exclusive.

Q`5`. Can an acute angle be negative?

Answer: No, an acute angle cannot be negative as it represents a measure of rotation less than `90°` and greater than `0°`.